A survey of failure criteria for brittle and ductile materials is presented. The maximum principal stress and the maximum principal strain criterion for brittle materials are introduced. The Tresca maximum shear stress and the von Mises energy criterion for ductile materials are formulated and applied to study the onset of plastic yield in thin-walled tubes and other structural members. The Mohr failure criterion is based on the consideration of Mohr's circles. The Coulomb–Mohr criterion for geomaterials incorporates the normal and shear stress, and the coefficient of internal friction. According to Drucker–Prager’s criterion, plastic yield occurs when the shear stress on octahedral planes overcomes the cohesive and frictional resistance to sliding. The fracture mechanics based failure criterion takes into account the presence of cracks. Failure occurs if the release of potential energy accompanying the crack growth is sufficient to supply the increase of the surface energy of expanded crack faces. The fracture criterion is also formulated in terms of the stress intensity factor K, whose critical value is the fracture toughness of the material.

The elastic behaviour of long and short fibre composites is described in Chapters 4–6. This involves considering the stresses in individual plies of a laminate (under an external load) and stress distributions within and around short fibres. This information is now used to explore how a composite material suffers microstructural damage, potentially leading to ultimate failure of some sort. There are two distinct aspects to these (highly important) characteristics. First, there is the onset and development of microstructural damage (mainly cracking of various types) as a function of applied load. Second, there are the processes that cause absorption of energy within a composite material as it undergoes such failure and fracture. The latter determine the toughness of the material and are treated on a fracture mechanics basis in Chapter 9. In the present chapter, attention is concentrated on predicting how applied stresses create stress distributions within the composite and how these lead to damage and failure. The treatment is largely oriented towards long fibre composites (particularly laminates), and also towards polymer-based composites, although most of the principles apply equally to discontinuous reinforcement and other types of matrix.